Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
242. The Diagonals' Balance · always bisect each other
The intersection point O is always the exact midpoint of both diagonals AC and BD.
In a parallelogram the diagonals bisect each other — they cross at a point O that is the midpoint of both AC and BD. A diagonal splits the figure into congruent triangles, which forces O to the middle of each diagonal.
What this lesson covers
Try to break it
Drag A, B, or C. D snaps to whatever position keeps ABCD a parallelogram, and O always lands at the midpoint of both diagonals. Try to drag a vertex so O drifts off either midpoint — impossible.
How you build it
Build a parallelogram and draw its diagonals.
- Point tool: mark point A at the bottom-left.
- Point tool: mark point B to the right of A.
- Point tool: mark point D above A — the top-left corner.
- Segment tool: join A to B.
- Segment tool: join A to D.
- Parallel tool: click D, then click on AB — the line through D is parallel to AB.
- Parallel tool: click B, then click on AD — the line through B is parallel to AD.
- Point tool: mark point C where the two parallel lines cross — this completes parallelogram ABCD.
- Segment tool: join A to C — one diagonal.
- Segment tool: join B to D — the other diagonal.
- Point tool: mark O where diagonals AC and BD cross — it lands exactly at the midpoint of both.
The proof, step by step
Prove that the diagonals of a parallelogram bisect each other.
- AB = DC (Opposite sides of a parallelogram are equal)
- ∠1 = ∠2 and ∠3 = ∠4 (Alternate angles)
- ΔAOB ≅ ΔCOD (ASA Congruence Criterion)
- OA = OC and OB = OD (CPCT)
Worked example
In a parallelogram ABCD, diagonals intersect at O. If AC = 24 cm and BD = 18 cm, what are the lengths of OA and OB?
Diagonals of a parallelogram bisect each other. Therefore, OA = AC/2 = 12 cm and OB = BD/2 = 9 cm.
- 12 cm, 9 cm — correct
- 12 cm, 18 cm
- 24 cm, 9 cm
- 6 cm, 9 cm